PDF Function


Computes probability density (mass) functions

Category: Probability

Alias: PMF

Syntax

PDF ( 'dist',quantile,parm-1, ,parm-k )

Arguments

'dist'

  • is a character string that identifies the distribution. Valid distributions are as follows :

    Distribution

    Argument

    Bernoulli

    'BERNOULLI'

    Beta

    'BETA'

    Binomial

    'BINOMIAL'

    Cauchy

    'CAUCHY'

    Chi-Square

    'CHISQUARE'

    Exponential

    'EXPONENTIAL'

    F

    'F'

    Gamma

    'GAMMA'

    Geometric

    'GEOMETRIC'

    Hypergeometric

    'HYPERGEOMETRIC'

    Laplace

    'LAPLACE'

    Logistic

    'LOGISTIC'

    Lognormal

    'LOGNORMAL'

    Negative binomial

    'NEGBINOMIAL'

    Normal

    'NORMAL''GAUSS'

    Normal mixture

    'NORMALMIX'

    Pareto

    'PARETO'

    Poisson

    'POISSON'

    T

    'T'

    Uniform

    'UNIFORM'

    Wald (inverse Gaussian)

    'WALD''IGAUSS'

    Weibull

    'WEIBULL'

  • Note: Except for T, F, and NORMALMIX, you can minimally identify any distribution by its first four characters .

quantile

  • is a numeric random variable.

parm-1, ,parm-k

  • are optional shape , location , or scale parameters that are appropriate for the specific distribution.

  • See: 'Details' on page 696 for complete information about these parameters

Details

Bernoulli Distribution

  • PDF ('BERNOULLI', x , p )

where

  • x

    • is a numeric random variable.

  • p

    • is a numeric probability of success.

    • Range: p 1

The PDF function for the Bernoulli distribution returns the probability density function of a Bernoulli distribution, with probability of success equal to p . The PDF function is evaluated at the value x . The equation follows:

click to expand

Note: There are no location or scale parameters for this distribution.

Beta Distribution

PDF ('BETA', x , a , b <, l , r >)

where

  • x

    • is a numeric random variable.

  • a

    • is a numeric shape parameter.

    • Range: a > 0

  • b

    • is a numeric shape parameter.

    • Range: b > 0

  • l

    • is the numeric left location parameter.

    • Default:

  • r

    • is the right location parameter.

    • Default:

    • Range: r > l

The PDF function for the beta distribution returns the probability density function of a beta distribution, with shape parameters a and b . The PDF function is evaluated at the value x . The equation follows:

click to expand

Note: The quantity is forced to be click to expand

Binomial Distribution

PDF ('BINOMIAL', m , p , n )

where

  • m

    • is an integer random variable that counts the number of successes.

    • Range: m = 0, 1,

  • p

    • is a numeric probability of success.

    • Range: p 1

  • n

    • is an integer parameter that counts the number of independent Bernoulli trials.

    • Range: n = 0, 1,

The PDF function for the binomial distribution returns the probability density function of a binomial distribution, with parameters p and n , which is evaluated at the value m . The equation follows:

click to expand

Note: There are no location or scale parameters for the binomial distribution.

Cauchy Distribution

  • PDF ('CAUCHY', x <, , » >)

where

  • x

    • is a numeric random variable.

    • is a numeric location parameter.

    • Default:

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the Cauchy distribution returns the probability density function of a Cauchy distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Chi-Square Distribution

  • PDF ('CHISQUARE', x , df <, nc >)

where

  • x

    • is a numeric random variable.

  • df

    • is a numeric degrees of freedom parameter.

    • Range: df > 0

  • nc

    • is an optional numeric non- centrality parameter.

    • Range: nc

The PDF function for the chi-square distribution returns the probability density function of a chi-square distribution, with df degrees of freedom and non-centrality parameter nc . The PDF function is evaluated at the value x . This function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central chi-square distribution. The following equation describes the PDF function of the chi-square distribution,

click to expand

where p c (.,.) denotes the density from the central chi-square distribution:

click to expand

and where p g ( y , b ) is the density from the gamma distribution, which is given by

click to expand

Exponential Distribution

  • PDF ('EXPONENTIAL', x <, » >)

where

  • x

    • is a numeric random variable.

  • »

    • is a scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the exponential distribution returns the probability density function of an exponential distribution, with the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

F Distribution

  • PDF ('F', x , ndf , ddf <, nc >)

where

  • x

    • is a numeric random variable.

  • ndf

    • is a numeric numerator degrees of freedom parameter.

    • Range: ndf > 0

  • ddf

    • is a numeric denominator degrees of freedom parameter.

    • Range: ddf > 0

  • nc

    • is a numeric non-centrality parameter.

    • Range: nc

The PDF function for the F distribution returns the probability density function of an F distribution, with ndf numerator degrees of freedom, ddf denominator degrees of freedom, and non-centrality parameter nc . The PDF function is evaluated at the value x . This PDF function accepts non-integer degrees of freedom for ndf and ddf . If nc is omitted or equal to zero, the value returned is from a central F distribution. The following equation describes the PDF function of the F distribution,

click to expand

where p f ( f , u 1 , u 2 ) is the density from the central F distribution with

click to expand

and where p B ( x , a , b ) is the density from the standard beta distribution.

Note: There are no location or scale parameters for the F distribution.

Gamma Distribution

  • PDF ('GAMMA', x , a <, » >)

where

  • x

    • is a numeric random variable.

  • a

    • is a numeric shape parameter.

    • Range: a > 0

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the gamma distribution returns the probability density function of a gamma distribution, with the shape parameter a and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Geometric Distribution

  • PDF ('GEOMETRIC', m,p )

where

  • m

    • is a numeric random variable that denotes the number of failures.

    • Range: m

  • p

    • is a numeric probability of success.

    • Range: p 1

The PDF function for the geometric distribution returns the probability density function of a geometric distribution, with parameter p . The PDF function is evaluated at the value m . The equation follows:

click to expand

Note: There are no location or scale parameters for this distribution.

Hypergeometric Distribution

  • CDF ('HYPER', x,N,R,n <, o >)

where

  • x

    • is an integer random variable.

  • N

    • is an integer population size parameter.

    • Range: N = 1, 2, ...

  • R

    • is an integer number of items in the category of interest.

    • Range: R = 0, 1, ..., N

  • n

    • is an integer sample size parameter.

    • Range: n = 1, 2, ..., N

  • o

    • is an optional numeric odds ratio parameter.

    • Range: o > 0

The PDF function for the hypergeometric distribution returns the probability density function of an extended hypergeometric distribution, with population size N , number of items R , sample size n , and odds ratio o . The PDF function is evaluated at the value x . If o is omitted or equal to 1, the value returned is from the usual hypergeometric distribution. The equation follows:

click to expand

Laplace Distribution

  • PDF ('LAPLACE', x <, , » >)

where

  • x

    • is a numeric random variable.

    • is a numeric location parameter.

    • Default:

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the Laplace distribution returns the probability density function of the Laplace distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Logistic Distribution

  • PDF ('LOGISTIC', x <, , » >)

where

  • x

    • is a numeric random variable.

    • is a numeric location parameter.

    • Default:

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the logistic distribution returns the probability density function of a logistic distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Lognormal Distribution

  • PDF ('LOGNORMAL', x <, , » >)

where

  • x

    • is a numeric random variable.

    • is a numeric location parameter.

    • Default:

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the lognormal distribution returns the probability density function of a lognormal distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Negative Binomial Distribution

  • PDF ('NEGBINOMIAL', m,p,n )

where

  • m

    • is a positive integer random variable that counts the number of failures.

    • Range: m = 0, 1, ...

  • p

    • is a numeric probability of success.

    • Range: p 1

  • n

    • is an integer parameter that counts the number of successes.

    • Range: n = 1, 2, ...

The PDF function for the negative binomial distribution returns the probability density function of a negative binomial distribution, with probability of success p and number of successes n . The PDF function is evaluated at the value m . The equation follows:

click to expand

Note: There are no location or scale parameters for the negative binomial distribution.

Normal Distribution

  • PDF ('NORMAL', x <, , » >)

where

  • x

    • is a numeric random variable.

    • is a numeric location parameter.

    • Default:

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the normal distribution returns the probability density function of a normal distribution, with the location parameter and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Normal Mixture Distribution

  • CDF ('NORMALMIX', x,n,p,m,s )

where

  • x

    • is a numeric random variable.

  • n

    • is the integer number of mixtures.

    • Range: n = 1, 2, ...

  • p

    • is the n proportions , p 1 , p 2 ,..., p n , where .

    • Range: p = 0, 1, ...

  • m

    • is the n means m 1 , m 2 ,..., m n .

  • s

    • is the n standard deviations s 1 , s 2 ,..., s n .

    • Range: s > 0

The PDF function for the normal mixture distribution returns the probability that an observation from a mixture of normal distribution is less than or equal to x . The equation follows:

click to expand

Note: There are no location or scale parameters for the normal mixture distribution.

Pareto Distribution

  • PDF ('PARETO', x,a <, k >)

where

  • x

    • is a numeric random variable.

  • a

    • is a numeric shape parameter.

    • Range: a > 0

  • k

    • is a numeric scale parameter.

    • Default: 1

    • Range: k > 0

The PDF function for the Pareto distribution returns the probability density function of a Pareto distribution, with the shape parameter a and the scale parameter k . The PDF function is evaluated at the value x . The equation follows:

click to expand

Poisson Distribution

  • PDF ('POISSON', n,m )

where

  • n

    • is an integer random variable.

    • Range: n =0,1,...

  • m

    • is a numeric mean parameter.

    • Range: m > 0

The PDF function for the Poisson distribution returns the probability density function of a Poisson distribution, with mean m . The PDF function is evaluated at the value n . The equation follows:

click to expand

Note: There are no location or scale parameters for the Poisson distribution.

T Distribution

  • PDF ('T', t,df <, nc >)

where

  • t

    • is a numeric random variable.

  • df

    • is a numeric degrees of freedom parameter.

    • Range: df > 0

  • nc

    • is an optional numeric non-centrality parameter.

The PDF function for the T distribution returns the probability density function of a T distribution, with degrees of freedom df and non-centrality parameter nc . The PDF function is evaluated at the value x . This PDF function accepts non-integer degrees of freedom. If nc is omitted or equal to zero, the value returned is from the central T distribution. The equation follows:

click to expand

Note: There are no location or scale parameters for the T distribution.

Uniform Distribution

  • PDF ('UNIFORM', x <, l,r >)

where

  • x

    • is a numeric random variable.

  • l

    • is the numeric left location parameter.

    • Default:

  • r

    • is the numeric right location parameter.

    • Default: 1

    • Range: r > l

The PDF function for the uniform distribution returns the probability density function of a uniform distribution, with the left location parameter l and the right location parameter r . The PDF function is evaluated at the value x . The equation follows:

click to expand

Wald (Inverse Gaussian) Distribution

  • PDF ('WALD', x,d )

  • PDF ('IGAUSS', x,d )

where

  • x

    • is a numeric random variable.

  • d

    • is a numeric shape parameter.

    • Range: d > 0

The PDF function for the Wald distribution returns the probability density function of a Wald distribution, with shape parameter d , which is evaluated at the value x . The equation follows:

click to expand

Note: There are no location or scale parameters for the Wald distribution.

Weibull Distribution

  • PDF ('WEIBULL', x,a <, » >)

where

  • x

    • is a numeric random variable.

  • a

    • is a numeric shape parameter.

    • Range: a > 0

  • »

    • is a numeric scale parameter.

    • Default: 1

    • Range: » > 0

The PDF function for the Weibull distribution returns the probability density function of a Weibull distribution, with the shape parameter a and the scale parameter » . The PDF function is evaluated at the value x . The equation follows:

click to expand

Examples

SAS Statements

Results

y=pdf( ' BERN ' ,0,.25);

0.75

y=pdf( ' BERN ' ,1,.25);

0.25

y=pdf( ' BETA ' ,0.2,3,4);

1.2288

y=pdf( ' BINOM ' ,4,.5,10);

0.20508

y=pdf( ' CAUCHY ' ,2);

0.063662

y=pdf( ' CHISQ ' ,11.264,11);

0.081686

y=pdf( ' EXPO ' ,1);

0.36788

y=pdf( ' F ' ,3.32,2,3);

0.054027

y=pdf( ' GAMMA ' ,1,3);

0.18394

y=pdf( ' HYPER ' ,2,200,50,10);

0.28685

y=pdf( ' LAPLACE ' ,1);

0.18394

y=pdf( ' LOGISTIC ' ,1);

0.19661

y=pdf( ' LOGNORMAL ' ,1);

0.39894

y=pdf( ' NEGB ' ,1,.5,2);

0.25

y=pdf( ' NORMAL ' ,1.96);

0.058441

  y=pdf('NORMALMIX',2.3,3,.33,.33,.34,   .5,1.5,2.5,.79,1.6,4.3);  

0.1166

y=pdf( ' PARETO ' ,1,1);

1

y=pdf( ' POISSON ' ,2,1);

0.18394

y=pdf( ' T ' ,.9,5);

0.24194

y=pdf( ' UNIFORM ' ,0.25);

1

y=pdf( ' WALD ' ,1,2);

0.56419

y=pdf( ' WEIBULL ' ,1,2);

0.73576




SAS 9.1 Language Reference Dictionary, Volumes 1, 2 and 3
SAS 9.1 Language Reference Dictionary, Volumes 1, 2 and 3
ISBN: N/A
EAN: N/A
Year: 2004
Pages: 704

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